An Extremum Solution of the Monin-Obukhov Similarity
Equations
The MIT Faculty has made this article openly available. Please share
how this access benefits you. Your story matters.
Citation
Wang, Jingfeng, and Rafael L Bras. “An Extremum Solution of
the Monin–Obukhov Similarity Equations.” Journal of the
Atmospheric Sciences 67.2 (2010): 485-499. © 2010 American
Meteorological Society
As Published
http://dx.doi.org/10.1175/2009jas3117.1
Publisher
American Meteorological Society
Version
Final published version
Accessed
Thu May 26 09:53:34 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/60345
Terms of Use
Article is made available in accordance with the publisher's policy
and may be subject to US copyright law. Please refer to the
publisher's site for terms of use.
Detailed Terms
FEBRUARY 2010
WANG AND BRAS
485
An Extremum Solution of the Monin–Obukhov Similarity Equations
JINGFENG WANG*
Massachusetts Institute of Technology, Cambridge, Massachusetts
RAFAEL L. BRAS
University of California, Irvine, Irvine, California
(Manuscript received 26 February 2009, in final form 14 July 2009)
ABSTRACT
An extremum hypothesis of turbulent transport in the atmospheric surface layer is postulated. The hypothesis has led to a unique solution of Monin–Obukhov similarity equations in terms of simple expressions
linking shear stress (momentum flux) and heat flux to mean wind shear and temperature gradient. The extremum solution is consistent with the well-known asymptotic properties of the surface layer. Validation of
the extremum solution has been made by comparison to field measurements of momentum and heat fluxes.
Furthermore, a modeling test of predicting surface heat fluxes using the results of this work is presented. A
critical reexamination of the interpretation of the Obukhov length is given.
1. Introduction
The Monin–Obukhov similarity theory (MOST)
(Obukhov 1946; Monin and Obukhov 1954) is considered
the most successful theory of atmospheric turbulence,
leading to innumerable publications over five decades of
active development and application. However, as Obukhov
cautioned in his seminal paper, MOST is semiempirical.
‘‘Empirical’’ is understood to imply that the mean wind
and temperature profiles are formulated (or fitted) based
on dimensional analysis (Bridgman 1931) rather than
derived from more fundamental physical laws (underlying the dynamics of turbulence); ‘‘semi’’ indicates
that some physics of turbulence does enter the formalism
through the selection of the key dimensional parameters.
Although not a physical law per se, MOST elements may
have physical interpretation. For example, the Obukhov
length scale, the most important parameter in the MOST,
was interpreted by Monin and Obukhov as the ‘‘thickness
of the sublayer of dynamic turbulence.’’
* Current affiliation: University of California, Irvine, Irvine,
California.
Corresponding author address: Jingfeng Wang, Department of
Civil and Environmental Engineering, University of California,
Irvine, E2167 Engineering Gateway, Irvine, CA 92697.
E-mail: jingfenw@uci.edu
DOI: 10.1175/2009JAS3117.1
Ó 2010 American Meteorological Society
In the classical MOST formalism, four dimensionless
state variables (i.e., momentum flux, heat flux, mean
wind shear, and temperature gradient) are expressed
as functions of a single dimensionless independent variable (i.e., vertical coordinate) and related through
two dimensionless governing equations according to the
Buckingham theorem (Buckingham 1914). Therefore, any
two of the state variables are expected to be uniquely
determined from the other two state variables. For instance, the heat flux can be determined by specifying the
temperature gradient and momentum flux. It turns out
that given wind shear and heat flux under stable conditions
do not always correspond to a unique value of momentum
flux, and neither do temperature gradient and heat flux
under unstable conditions always yield a unique value of
momentum flux according to the widely used MOST
equations. There are no known rules for selecting a
unique physical solution when that happens. Since the
two governing equations include two nonlinear functions characterizing the atmospheric instability, the heat
flux must be obtained numerically using an iterative
procedure. In practice, we sometimes face the problem
of a nonconverging iteration. An arbitrary decision has
to be made (to stop the nonconverging iterations) when
this situation occurs, resulting in inaccurate, even potentially incorrect, solution(s). One main motivation of
this study is to find a rationale leading to a unique solution to avoid the iterative procedure in the theoretical
486
JOURNAL OF THE ATMOSPHERIC SCIENCES
and modeling applications of the MOST where the
parameterization of eddy diffusivity becomes much
simplified. We explore the possibility of an extremum
principle.
Basic physical laws, such as Newton’s law in classical
mechanics, are often equivalent to extremum principles
(e.g., Lanczos 1970). Busse (1978) developed an optimality theory of turbulence with some success. Whether
there exists general simple extremum principle(s) for
turbulence is unclear; nevertheless, extremum hypotheses have been proposed. Cheng et al. (2005) argued
that the MOST holds once the turbulence ‘‘became
homogeneous and stationary’’ when the wind shear and
buoyancy are in equilibrium. It is well known that some
physical quantities are extremized for a system at equilibrium. The principle of minimum potential energy in
dynamics (Goodman and Warner 2001) and the second
law in thermodynamics (Kondepudi and Prigogine 1998)
are two good examples. The clue pointing to the proposed
hypothesis comes from the fact that the multiple solutions
allowed by the MOST equations become unique when
some of the state variables reach extrema, referred to
herein as the extremum solution of the Monin–Obukhov
similarity equations. The proposed extremum hypothesis
may be justified qualitatively with the emerging theory
of maximum entropy production (MEP) (Dewar 2003,
2005). Theoretical justification of the hypothesis will
also be offered by confirming that the extremum solution has the desired asymptotic properties. We present
two kinds of experimental evidence in support of the
hypothesis: direct validation of a newly derived extremum solution (i.e., a relationship between heat flux and
momentum flux) and heat fluxes predicted by a model of
surface energy balance based on the MEP theory where
the proposed extremum solution is used.
2. An extremum hypothesis
The idea of the proposed hypothesis results from
reasoning based on some physical and mathematical
arguments. Based on daily experience, it is reasonable to
state that mechanical mixing (forced convection) is arguably more effective than buoyancy (free convection)
in transporting heat and momentum; ‘‘effectiveness’’
may be measured by the relative magnitude of fluxes to
the corresponding scalar gradients. The effectiveness of
a transport mechanism implies that the corresponding
flux(es) would be extremum (under certain conditions
or constraints). In modeling turbulence in the atmospheric surface layer (ASL) using the MOST, we expect
to obtain a unique solution of unknown quantities (such
as fluxes of momentum and heat), given the input information (such as temperature gradient and wind shear).
VOLUME 67
By inspection of the similarity equations of the MOST
(see below), we realized that unique solution of the
similarity equations coincides with extremum fluxes.
The exact conditions under which this (i.e., extremum
and uniqueness) occurs may be expressed in terms of the
following extremum hypothesis about the turbulent
structure of the ASL described by the MOST:
d
Within the atmospheric surface layer in an environment
for which the MOST applies, momentum flux would
reach such values as to minimize heat flux and wind
shear under stable conditions and to minimize heat flux
and temperature gradient under unstable conditions.
Here we follow the convention in hydrometeorology
that a flux is positive when it points away from the land
surface and an entity increasing with height is defined as
positive. Note that minimizing a negative variable is
equivalent to maximizing its absolute value. According
to this definition, minimum (downward) heat flux under
stable conditions implies its magnitude is maximum, and
minimum (negative) temperature gradient under unstable condition implies its magnitude being maximum.
Some clarification is needed to avoid ambiguity and
confusion. Turbulence is a dynamic system with a large
number of degrees of freedom. There are many possible
scenarios of turbulent flow that conserve mass, momentum, and energy and have other physical and/or mathematical constraints. For the stationary and homogeneous
turbulence in the ASL, a variety of combinations of
fluxes and gradients described by the (two) dimensionless equations in the MOST are physically possible.
Theoretically, there are an infinite number of combinations of the four unknowns (i.e., wind shear, temperature
gradient, heat flux, and momentum flux) that satisfy the
two dimensionless equations. The proposed extremum
principle selects the unique extremum solution among
the possible ones allowed by the dimensionless equations
in the MOST. The physical significance of the extremum
solution will be further elaborated below.
3. Extremum solution
In the framework of the MOST, the turbulence in the
ASL is characterized by four independent variables: the
distance from the surface z, shear stress t (equivalent to
a velocity scale u*), kinematic heat flux H/rCp (equivalent to a temperature scale u*), and the buoyancy g/T0
related through
t
5 u2* ;
r
H
5 u* u* ,
rcp
(1)
(2)
FEBRUARY 2010
where r is the (constant) density of air, cp is the heat
capacity of the air at constant pressure, and T0 is a reference temperature. The well-known Obukhov length L
is defined as
,
g H
3
L 5 u* k
,
(3)
T 0 rcp
where g is the gravitational acceleration and k is the von
Kármán constant.
According to the Buckingham p theorem (Buckingham
1914), dimensionless wind shear and temperature gradient may be expressed in terms of a dimensionless composition of the independent variables
z
kzU z
5 fm
;
L
u*
(4)
z
kzQz
,
5 fh
L
u*
(5)
where the subscript z of U and Q stands for partial derivative with respect to z. In Eqs. (4) and (5) fm and fh are
empirical functions introduced by Monin and Obukhov to
represent the effect of the stability on the mean profiles
of wind speed and temperature. The most popular
functional forms of fm and fh (used in this study) were
proposed by Businger et al. (1971):
fm 5 1 1 b
z
;
L
(6)
fh 5 a 1 b
z
L
(7)
under stable conditions, L . 0, and
z 1/4
fm 5 1 g1
;
L
z 1/2
fh 5 a 1 g2
L
(8)
a. Stable layer
The wind profile described by Eqs. (4) and (6) leads to
two equivalent expressions for H and Uz,
H
1 T0 2
u (u kzU z );
5
rCp
b gkz * *
u
gkz 1 H
Uz 5 * 1 b
kz
T 0 u3 rCp
*
(10)
!
(11)
as functions of u*. These relationships are illustrated
in Figs. 1a and 1b. Here H and Uz are not monotonic
functions of u*, opening the possibility of an extremum
hypothesis applying to the wind profile to yield a unique
solution.
The temperature profile described by Eqs. (5) and (7)
leads to two equivalent expressions,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !
4b gk2 z2 Qz
;
11 2
a T 0 u2
*
!
g H 1 H 1
a T0
Qz 5 b
:
T 0 rcp u* rcp u3 b gkz
*
H
a T0 3
u 1
5
rCp
2b gkz *
(12)
(13)
H and Qz as functions of u* are illustrated in Figs. 2a and
2b and indicate that H and Qz are monotonic functions
of u*, assuring a unique solution. No extremum hypothesis would be associated with the temperature profile.
As shown in Fig. 1, a given value of Uz (H) leads to
two solutions of u* for a given H (Uz) except when Uz
(H) reaches its minimum. It is straightforward to verify
that H in Eq. (10) is minimized (or 2H is maximized)
when
Uz 5
3 u*
2 kz
(14)
and Uz in Eq. (11) is minimized when
(9)
under unstable conditions, L , 0. The universal empirical constants in the above expressions are estimated as
a ; 0.75,
487
WANG AND BRAS
b ; 4.7, g1 ; 15, g2 ; 9,
where the approximation symbol emphasizes that these
values have appreciable uncertainties. Alternative forms
of fm and fh have been suggested by other authors (e.g.,
Beljaars and Holtslag 1991). They do not differ qualitatively from those of Businger et al. and will not affect our
analysis below.
3
H
1 T 0 u*
.
5
rcp
2b g kz
(15)
Since Qz is a monotonic function of H and u* according to Eq. (13), substituting H in Eq. (15) into Eq. (13)
leads to
1 1 T 0 u* 2
.
Qz 5 a 1
2 2b g kz
(16)
Equations (14)–(16) are the extremum solution under
stable condition.
488
JOURNAL OF THE ATMOSPHERIC SCIENCES
FIG. 1. Plot of (a) H vs u* according to Eq. (10) for a given Uz and
(b) Uz vs u* according to Eq. (11) for a given H in a stable layer.
The dashed line represents the wind profile in a neutral layer. The
graph is for illustration purposes and is not drawn to scale and has
arbitrary units.
Substituting the extremum solution of Uz given
in Eq. (14) into Eq. (11) leads to Eq. (15), while
substituting the extremum solution of H in Eq. (15)
into Eq. (10) leads to Eq. (14). This is not surprising
since Eqs. (10) and (11) are identical. This consistency
implies that the momentum flux that minimizes heat
flux also minimizes wind shear. Therefore, a minimum
(negative) heat flux coincides with a minimum wind
shear. The fact that the extremum solution of u* is
VOLUME 67
FIG. 2. As in Fig. 1 but of (a) H vs u* according to Eq. (12) for
a given Uz and (b) Qz vs u* according to Eq. (13) for a given H in
a stable layer.
determined by the wind profile suggests that dynamic
mixing is responsible for the heat transfer in a stable
layer where thermal convection is suppressed. The
extremum hypothesis implies that dynamic mixing not
only dominates the heat transfer, but also does so effectively. The kinetic energy drawn from the mean flow
is used in such an efficient way (with a minimum wind
shear) that the atmosphere takes an optimal path toward a potential thermal equilibrium by maximizing
the heat flux.
FEBRUARY 2010
WANG AND BRAS
489
b. Unstable layer
The wind profile under unstable condition described
by Eqs. (4) and (8) has two equivalent expressions,
2
3
!4
u
T
H
1 0 34
*
u
5
15;
rCp
g1 gkz * kzU z
u7/4
gkz H
U z 5 * u3* 1 g1
T 0 rCp
kz
(17)
!1/4
:
(18)
H and Uz, given the other, in Eqs. (17) and (18) are illustrated in Figs. 3a and 3b. Contrary to the case of
a stable layer, H and Uz are monotonic functions of u*.
The temperature profile described by Eqs. (5) and (9)
can also be written as two equivalent forms,
!2
H
1 T 0 3 g2 gk2 z2 Qz
u
5
rCp
2g2 gkz * a T 0 u2
*
2
3
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!2ffi
u
2
u
a T 0 u*
6
7
341 1 t1 1
5;
g2 gk2 z2 Qz
1/2
H u*
Qz 5 a
rCp kz
gkz H
u* 1 g2
T 0 rCp
(19)
!1/2
3
:
(20)
H and Qz as functions of u* are illustrated in Figs. 4a and
4b. Contrary to the case of stable layer, H and Qz are not
monotonic functions of u*, opening the possibility of an
extremum hypothesis applying to the temperature profile to yield a unique solution.
As shown in Fig. 4, a certain Qz (H) corresponds to two
values of u* unless Qz (H) reaches a minimum to have
a unique solution of u*. It is straightforward to show that
minimizing Qz (or maximizing Qz) in Eq. (20) leads to
3
H
2 T 0 u*
.
5
rcp
g 2 g kz
(21)
Meanwhile, minimizing H in Eq. (19) leads to
2 a T 0 u*
Qz 5 pffiffiffi
3 g2 g kz
2
.
(22)
Since Uz is a monotonic function of H and u* according to
Eq. (18), substituting H in Eq. (21) into Eq. (18) leads to
Uz 5
g2
g2 1 2g1
1/4
u*
.
kz
(23)
FIG. 3. As in Fig. 1 but of (a) H vs u* according to Eq. (17) for
a given Uz and (b) Uz vs u* according to Eq. (18) for a given H in an
unstable layer.
Equations (21)–(23) are the extremum solution under
unstable condition.
The extremum solutions under unstable condition are
also consistent. Substituting the extremum solution of
Qz in Eq. (22) into Eq. (19) leads to Eq. (21), while
substituting the extremum solution of H in Eq. (21) into
Eq. (20) leads to Eq. (22). This is again expected since
Eq. (19) and Eq. (20) are identical. The consistency implies that momentum flux that minimizes heat flux also
minimizes temperature gradient. Therefore, a minimum
490
JOURNAL OF THE ATMOSPHERIC SCIENCES
VOLUME 67
temperature profile instead of wind shear suggests that
buoyancy is what limits the heat transfer in an unstable
layer. It also implies that thermally driven convection is
less efficient than mechanical mixing as a heat transfer
mechanism because the maximum temperature gradient
corresponds to a minimum heat flux. It is true that heat
flux in an unstable layer is usually higher than that in
a stable layer. However, this is due to the greater daytime heat supply at the surface rather than the effectiveness of buoyancy-driven turbulent transport.
c. Neutral layer
The wind and temperature profiles under neutral condition have a trivial solution,
H 5 0,
(24)
Qz 5 0,
(25)
u
Uz 5 * .
kz
(26)
It is important to point out that Eq. (26) results directly from the dimensional analysis, although it can also
be formally derived from Eq. (4) as H / 0. Theoretically, L does not exist for a neutral layer even though
L / ‘ as H / 0 according to Eq. (3). Based on the
dimensional argument, the only length scale for the case
of a neutral layer is z. Hence, L is not continuous in H at
H 5 0; that is,
lim L 5 ‘ 6¼ L(H 5 0),
H!0
(27)
where L on the right-hand side is understood as a length
scale, if it exists. This intricate point will be elaborated
further in section 5.
FIG. 4. As in Fig. 1 but of (a) H vs u* according to Eq. (19) for
a given Uz and (b) Qz vs u* according to Eq. (20) for a given H in an
unstable layer.
(positive) heat flux concurs with a minimum (negative)
temperature gradient. The solution of Qz in Eq. (22), as
well as that given in Eq. (16), appears to be what
Townsend (1962) called the ‘‘natural convection (solution)’’ as far as z dependence is concerned. In fact, all
three regimes of Qz according to Townsend—that is,
}z21, }z24/3, and }z22—can be obtained from the extremum solution. Contrary to the case of a stable layer,
the association of the extremum hypothesis with the
d. Eddy diffusivity for heat transfer
The extremum solution derived above allows H
to be formally expressed in terms of the temperature
gradient,
H
›Q
5 Ck kzu*
,
rcp
›z
(28)
where Ck is a constant,
8pffiffiffi
3
>
>
< ,
a
Ck 5
2
>
>
:
,
(1 1 2a)
unstable
(29)
stable.
FEBRUARY 2010
491
WANG AND BRAS
Then, the eddy diffusivity for heat transfer, KH, is parameterized as
KH 5 Ck kzu* ,
(30)
TABLE 1. A summary of the extremum solution based on the
Monin–Ohbukov similarity theory. The constants in the equations,
a ; 0.75 or 1, b ; 4.7, g1 ; 15, and g 2 ; 9, were reported by
Businger et al. (1971).
Stable
referred to as the extremum solution of the eddy diffusivity. Later in the paper we will show that KH in
Eq. (30) plays an important role in modeling the energy
balance over a land surface.
Uz
e. Properties of the extremum solution
H
rcp
The existence of the extremum solution (summarized
in Table 1) based on the MOST results from two (not
very restrictive) assumptions: 1) fm(z) and fh(z) are
well-behaved functions of z that do not even have to be
monotonic and 2) any two of the states variables (i.e.,
Uz, Qz, u*, and H) are uniquely determined given the
other two. The basic features of Uz, Qz, and H in terms of
u*, as shown in Figs. 1–4, are independent of the functional forms of fm and fh as long as they are well behaved in terms of z [ z/L.
A remarkable feature of the extremum solution is that
Uz, Qz, and H expressed in terms of u* and z have exactly the same functional forms for unstable and stable
conditions except for the different proportionality coefficients. This is somewhat surprising, but does make
qualitative sense. For example, with the same u*, heat
flux in an unstable layer is about twice as strong as in
a stable layer. Also for the same u*, the temperature
gradient in a stable layer is greater than that of an unstable layer, and wind shear in a stable layer is greater
than that of an unstable layer. These results do agree
with our experience and intuition. A second notable
feature is that heat flux is nonlinearly dependent on
temperature gradient. Eliminating u* in the expressions
of H and Qs leads to an explicit nonlinear equation
linking heat flux to temperature gradient. This is consistent with the classical flux-gradient equation in which
the nonlinearity is through the diffusivity parameter that
is a nonlinear function of H. The eddy diffusivity parameterized using the extremum solution (see section 3d)
is also a nonlinear function of H. An application of this
parameterization will be presented in section 4b(2), which
also serves as a validation of the extremum solution.
We note that the assumption of height invariant fluxes
of heat and momentum is not required in the formulation of the governing equations in the MOST unless
explicit analytical expressions of wind and temperature
distributions are desirable when integrating the dimensionless gradient equations, that is, Eqs. (4) and (5). The
derivation of the extremum solution does not require
that assumption either. As a result, the wind speed will
deviate from the log profile when u* varies with z ac-
Qz
3 u*
2 kz
1 1 T 0 u* 2
a1
2 2b g kz
3
1 T 0 u*
2b g kz
Unstable
1/4
u*
g2
g2 1 2g 1
kz
2 a T 0 u* 2
pffiffiffi
3 g 2 g kz
3
2 T 0 u*
g 2 g kz
Neutral
u*
kz
0
0
cording to the extremum solution, regardless of heat
flux. Hence, the extremum solution is general and not
limited to the case of height-independent fluxes of momentum and heat. Significant variation of heat flux with
height has been observed (Elliott 1964).
4. Validation of the extremum solutions
The extremum solution may be justified theoretically
and experimentally. The theoretical justification has two
components: 1) the extremum solution is consistent with
the recent development in nonequilibrium thermodynamics, that is, the emerging theory of maximum entropy production, and 2) the extremum solution has the
correct asymptotic properties. The experimental justification comes from two tests: 1) direct confirmation of
the extremum solution using field observations and 2)
successful prediction of land surface energy budget by
a model formulated using the extremum solution.
a. Theoretical justification
1) NONEQUILIBRIUM THERMODYNAMICS
The recent advances in nonequilibrium thermodynamics, that is, the emerging theory of maximum entropy production (MEP) (Dewar 2003, 2005), offer
a new tool for characterizing and modeling atmospheric
turbulent transfer. The MEP theory is a derivative of the
theory of maximum entropy (MaxEnt) first formulated
as a general method to assign probability distribution
in statistical mechanics (Jaynes 1957). The theoretical
foundation of MaxEnt is Bayesian probability theory
(e.g., Jaynes 2003) in which the concept of entropy is
defined as a quantitative measure of information (Shannon and Weaver 1949) for any systems that need to be
described probabilistically for making statistical inferences. The MaxEnt distribution is interpreted as the
most probable and macroscopically reproducible state
among all physically possible states. In the formalism of
492
JOURNAL OF THE ATMOSPHERIC SCIENCES
MEP, the microscopic configurations of a nonequilibrium
system are characterized by a small number of observable
parameters such as macroscopic fluxes and the corresponding scalar gradients that are related functionally. It
remains uncertain at the moment whether the proposed
extremum hypothesis can be ‘‘proved’’ using the MEP
theory. Nonetheless, the extremum hypothesis appears to
be consistent with the MEP theory, implying that the
extremum hypothesis may result from some fundamental
laws of nonequilibrium thermodynamics, an ongoing research subject.
According to the MEP theory (Dewar 2005, p. L378),
imposed gradients of temperature and wind speed, for
example, would correspond to maximum (or minimum)
fluxes of heat and momentum and vice versa. Under
a stable condition, the dominant transport mechanism in
a boundary layer is forced convection driven by the synoptic wind where the effect of thermal forcing on the
transport is relatively weak. This is equivalent to the
situation of imposing a temperature gradient. Then
the MEP predicts that (downward) heat flux would be
maximized. This is the same property of the extremum
solution (Fig. 1a) where the actual friction velocity
corresponds to minimum (negative) heat flux. Under an
unstable condition, the turbulence in the ASL is driven
by combined free and forced convection. This is the
situation of imposed heat flux due to the fact that sensible heat flux into the atmosphere is determined by
maximum evaporation (Wang et al. 2004). Then MEP
predicts a maximum temperature gradient in agreement
with the extremum solution that corresponds to a maximum (inverse) temperature gradient (Fig. 4b). Several
recent studies (Ozawa et al. 2001; Lorenz and McKay
2003; Kleidon et al. 2006) have suggested that various
maximum transport properties of the atmospheric flows
may be explained by the MEP under specific boundary
conditions, although these studies did not directly use
the formalism of Dewar (2005). We acknowledge that
we have not been able to derive the entire extremum
solution directly using the MEP formalism. It is still
unclear whether the extremum solution of wind shear
and momentum flux agrees with the MEP prediction,
a subject of ongoing research. Nonetheless, the consistency of the extremum solution with the MEP theory
demonstrated here is the first step in that direction.
2) ASYMPTOTIC PROPERTIES
Here we demonstrate that the extremum solution has
the correct asymptotic properties of the MOST associated with a pure convective regime in which the mean
wind shear vanishes. These well-known asymptotic
properties have been obtained from various arguments
independent of those behind the extremum solution.
VOLUME 67
(i) Unstable layer
One situation of large z/L (for a fixed L) is the ‘‘purely
thermal turbulence with zero wind’’.1 Under the condition of zero (mean) wind, mechanical mixing plays no
role, hence u* drops out of the similarity equations in the
MOST. Eliminating u* in the extremum solution of H
and Qz (Table 1) leads to
!2/3 2a 2 1/3 H
g 1/3
Qz 5 pffiffiffi
(kz)4/3
rcp
T0
3 g2
C u z 1/3
5 h *
,
(31)
3 kz L
where u* is defined as in Eq. (2). Equation (31) is
identical to that obtained by Monin and Obukhov (1954)
[see Eqs. (2.182) and (2.184) in the English translation]
based on the argument that the purely thermal turbulence is self-similar, and Ch (unspecified in their paper) is
given as
pffiffiffi 2 1/3
Ch 5 2a 3
’ 2.0.
g2
(32)
The asymptotic extremum solution of Uz (Table 1)
can be expressed in two different forms. The first one is
an expression where u* remains2 in the equation,
U z 5 C1m u2*
H
rcp
!1/3 g
T0
1/3
u z 1/3
z4/3 5 C1m *
,
kz L
(33)
where the constant C1m is
C1m
g2
5
g2 1 2g 1
1/4 1/3
2
k4/3 ’ 1.4.
g2
(34)
This is the solution given in Monin and Obukhov
(1954) [Eqs. (2.180) and (2.185) in the English translation] as the limiting case of purely thermal turbulence.
However, it is inconsistent with the definition of the free
convection; that is, u* / 0. They did not explain why u*
vanishes in the case of purely thermal turbulence but
1
Monin and Obukhov (1954) defined the purely thermal turbulence as the case of zero wind and zero friction velocity. Theoretically, zero friction velocity only requires zero mean wind shear,
while wind speed is not necessarily zero.
2
In Monin and Obukhov (1954), the authors argued that wind
and temperature will have similar profiles, following the 21/ 3-power
law. Then u* will not drop out of the equation as required by the
definition of free convection, an obvious inconsistency in their
analysis.
FEBRUARY 2010
remains in the equation. Later, Kader and Yaglom
(1990) derived the same solution using a new argument
called ‘‘directional dimensional analysis.’’ Kader and
Yaglom showed that Eq. (33) is the wind profile of a
dynamic–convective layer using a three-sublayer model of
an unstable surface layer (dynamic, dynamic–convective,
and free convective).
The second form is obtained using the extremum solution of H and Uz to eliminate u*,
Uz 5
g 1/3 5
2
2
493
WANG AND BRAS
g2
g2 1 2g1
1/4
H
rcp
!1/3 g
T0
1/3
(kz)2/3
b. Experimental justification
C2m u* z 1/3
,
3 kz L
(35)
1/4
g 1/3 g
2
2
2
Cm 5 3
’ 3.4.
2
g2 1 2g1
(36)
Equation (35) is identical to the wind profile of the
free convective sublayer given by Kader and Yaglom
(1990) based on the directional dimensional analysis.
Curiously, Eq. (35) implies that there could be nonzero
wind shear in the case of purely thermal turbulence
where u* vanishes when L is fixed.
(ii) Stable layer
Turbulence would degenerate were it not for dynamic
mixing. Hence, u* remains important for a stable layer.
Monin and Obukhov argued that ‘‘turbulence characteristics must not explicitly depend on the distance z
from the underlying surface’’ for large z/L (for a fixed L)
in a stable layer. Eliminating z from the extremum solutions of H, Uz, and Qz leads to
g H 1
;
T 0 rcp u2
*
g H
Qz 5 (1 1 2a)b
T 0 rcp
(37)
!2
1
.
u4*
(38)
They are identical to those given by Monin and Obukhov
[1954, Eqs. (2.188) and (2.189)] with the stationary
Richardson number, R,
R5
1) DIRECT VALIDATION OF THE EXTREMUM
SOLUTION
where the constant C2m is
U z 5 3b
Obukhov result but also specifies the parameter that
they were unable to find.
When H and u* are independent of z as commonly
assumed for the ASL, Eq. (28) implies Qz } z21, which is
what Priestley (1955) and Taylor (1956) called the
‘‘forced convection regime.’’ Dyer (1964, p. 153) summarized four cases (i.e., near-neutral, Priestley regime,
and two Townsend regimes) in which Qz was expressed
in terms of various power laws of z, all of which can be
obtained from the extremum solutions. For example the
two Townsend regimes are identical according to the
extremum solution.
1
1
’ .
3b 14
(39)
Monin and Obukhov did not give its value except for
stating R to be no greater than the critical value Ricr. The
extremum solution not only retrieves the Monin and
Direct validation of the extremum solution requires
independent measurements of mean wind shear, temperature gradient, momentum, and heat flux. The fluxes
are routinely measured using eddy-covariance devices in
field campaigns. However, most field experiments do
not have multiple-level measurements of wind velocity
and temperature suitable for accurate retrieval of the
mean wind shear and temperature gradient. Here we
focus on testing the relationship of H ; u*3 predicted by
the extremum solution.
The data products from two field experiments are
used in this study: at Owens Lake, California, during
20 June–2 July 1993 and at Lucky Hill near Tombstone,
Arizona, during 2–17 June 2008. The Owens Lake site has
a bare soil surface, whereas the Lucky Hill site was covered with sparse dry shrubs. Eddy-covariance systems
were employed at both locations to measure turbulent
variables as well as soil variables including ground heat
fluxes and skin temperature. Further details about the
two experiments can be found in Katul (1994) and Wang
and Bras (2009).
The measured H versus u* at Owens Lake and Lucky
Hill are shown in Figs. 5 and 6, respectively. It is evident
that the data points tend to follow a straight line with
a 3:1 slope according to the extremum solution. Relatively large scatter in the H versus u* plots does not
allow accurate estimates of the coefficients in the regression equations of H 2 u*3 . Nonetheless, Figs. 5 and 6
indicate that they are comparable to the theoretical
value. The extremum solution is supported, at least
qualitatively, by the field observations.
Additional observational validation was given by Dyer
(1967) nearly forty years ago. Dyer estimated the dimensionless heat flux,
H* 5
H
H[rcp (g/T 0 )1/2 Q2z/3 z2 ],
(40)
494
JOURNAL OF THE ATMOSPHERIC SCIENCES
FIG. 5. Observed H vs u* for an unstable layer (H . 0) at the Owens
Lake site.
using field measurements. The estimated H* ranges
from 1.01 to 1.40 with a mean 1.15. According to the
extremum solution, H* is obtained as
!
g 1/2 pffiffi3ffi 3/2
2
’ 1.19,
H* 5 k
a
2
2
(41)
which is in good agreement with the observation. In
addition, Taylor (1959) also showed that H* } k2 with
values ranging from 0.76 to 2.28.
2) MODELING OF LAND SURFACE ENERGY
BUDGET
The extremum solution may be quantitatively validated through a model of energy balance over a dry land
surface based on the MEP theory (Wang and Bras 2009)
briefly described in the appendix. In the MEP model of
surface energy balance, the extreme of a so-called dissipation function of sensible and ground heat fluxes is
found under the constraint of conservation of energy.
One key component of the dissipation function is the
‘‘thermal inertia’’ of turbulent diffusion defined in
analogy to that for conduction where the molecular
diffusivity is replaced by eddy diffusivity KH. The extremum solution offers a mathematically more tractable
parameterization of KH expressed in Eq. (30). Compared to the common MOST-based parameterization of
the eddy diffusivity (e.g., Arya 1988, p. 161), the extremum solution allows KH to be expressed as an explicit
VOLUME 67
FIG. 6. Observed H vs u* for an unstable layer (H . 0) at the Lucky
Hill site.
nonlinear function of sensible heat flux alone. This new
parameterization of KH using the extremum solution
leads to a simple solution of sensible and ground heat
flux H and G [see Eqs. (A1) and (A2)]. Figure 7 compares H and G predicted by the MEP model with the
observed fluxes at the Owens Lake site. Close agreement is evident between the modeled and observed
fluxes under both stable (nighttime) and unstable (daytime) conditions.
The success of the MEP model in which the extremum
solution plays an important role clearly demonstrates
the potential applications of the extremum solution.
This case study is a validation of both the extremum
solution and the MEP theory, which have been developed independently so far. Furthermore, the MEP
model sheds light on the link between them as 1) the
extremum solution may be derived directly from the
MEP theory and 2) the correction prediction of heat
fluxes by the MEP model uses the extremum solution
while the basic idea of the MEP theory is independent of
the extremum solution. Therefore, this type of modeling
test, complementary to the direct confirmation of the
scaling relations as in Table 1 against measurements of
fluxes and mean gradients, offers strong support of the
extremum solution as well.
c. Implications of the extremum solution
The extremum solution suggests a possible simplification of the classical MOST formalism: reducing the
two empirical functions, fm and fh in Eqs. (4) and (5), to
FEBRUARY 2010
WANG AND BRAS
495
FIG. 7. The MEP model prediction (dashed) of (a) H and (b) G vs the observed (solid) fluxes
(observed Rn not shown). Data collected at Owens Lake, California, 20 Jun–2 Jul 1993.
some empirical constants as fm and fh are obtained
from curve fitting assuming constant momentum and
heat flux (hence constant L) in z. Note that z-invariant
flux profiles are only an approximation. This approximation is acceptable, maybe even necessary in practice,
but it has severe theoretical drawbacks: it violates
conservation laws and is inconsistent with the asymptotic properties of the surface layer. The extremum
solution corresponds to z 5 0.1, 20.2 (particularly for
larger z), around which the majority of the data points
cluster in z ; fm, fh empirical observations (see Businger et al. 1971). In addition, the assumption of constant
L makes the data points appear more variable than they
should on the z ; fm, fh diagrams, especially for larger
z. The above suggests the possibility of reducing two
empirical functions, fm and fh, to some empirical
constants in the formalism of the MOST. It may sound
like a circular argument as the derivation of the extre-
mum solution uses the prescribed fm and fh functions.
Yet, the existence of an extremum solution does not
depend on the functional forms of fm and fh. It is true
that the parameters in the extremum solution are related to those in the prescribed fm and fh functions
(i.e., a, b, g1, and g2 identified in Businger et al.). They
could be obtained by fitting the extremum solution (as in
Table 1) to field data directly. Use of the prescribed fm
and fh merely facilitates the derivation of the extremum solution. In fact, a number of previous studies
have hinted at this possibility, suggesting fm and fh
being constant for z/L $ ;0.5 (e.g., Handorf et al. 1999;
Johansson et al. 2001; Brutsaert 2005, p. 48). Constant
fm and fh for larger z implies that 1) the assumption
of constant L is only valid for a limited range of z (see
section 5) and 2) L would be proportional to z for larger
z, corresponding to a constant z, as predicted by the
extremum solution.
496
JOURNAL OF THE ATMOSPHERIC SCIENCES
The original MOST equations, (4) and (5), do not
uniquely determine the flux-gradient relationships, such
as that between H and u* illustrated in Fig. 1a, for almost
any nontrivial fm and fh functions. Such nonuniqueness
is more than a technical inconvenience. It signals potential loopholes in the classical treatment of the MOST
including violation of conservation laws and inconsistency in asymptotic properties. The extremum solution
completely removes this nonuniqueness. Uniqueness is
not only a technical advantage of the extremum solution
(e.g., to avoid the problem of nonconverging iteration)
but, more importantly, a desired property of the MOST
since the multiple-valued relationship resulting from
a mathematical artifact is physically unrealistic. Specifically for the case of the unstable layer shown in Fig. 4a,
the original MOST equations allow two values of u*
corresponding to a given H except at the bottom of the
curve. Yet, according to the definition of L in Eq. (3), u*
has only one real root for any given H regardless of L
being constant in, or varying with, z. Then, the extremum solution must be the only mathematically consistent and physically realistic solution among all possible
ones allowed by the MOST equations.
5. On the interpretation of Obukhov length
The asymptotic extremum solution under unstable
conditions presented in section 4.2 is known as the
‘‘1/ 3-power law’’, a character of the ‘‘free convection regime’’. Yet, it has been found that (Monin and Yaglom
1971, p. 482, 497), the ‘‘1/ 3-power law’’ holds throughout
nearly the entire surface layer; they wrote ‘‘. . . all existing
data show that the ‘1/ 3-power law’ (which refers theoretically only to the case of very large negative z3) begin
to be valid at unexpectedly small values of z of the order
of 20.1 (or even of several hundredth).’’ This phenomenon is attributed, according to Monin and Yaglom, to
the dominance of buoyancy-driven thermal turbulence
as they argued ‘‘. . . convection produces vertical turbulent mixing much more effectively than wind shear.
As a result, the thickness of the dynamic sub-layer in fact
comprises only a small part of jLj’’ ( p. 487). Kader and
Yaglom (1990) further argued that the length scale of
the dynamic sublayer is actually much smaller than the
original Obukhov length according to the directional
dimensional analysis. The above arguments imply that
the dynamic sublayer is unimportant and the entire
boundary layer is essentially thermally driven, assuming
L being independent of z. Yet, the Obukhov length L
has been interpreted as the thickness of the dynamic
3
Here z 5 z/L.
VOLUME 67
sublayer in which the thermal factor plays no significant
role, a concept first proposed by Obukhov (1946) and
promoted by Monin and Obukhov (1954) and Monin
and Yaglom (1971). The classical reasoning leading to
this view is that the limiting case of z 5 z/L / 0 corresponding to a neutral layer of H 5 0 is equivalent to the
limiting case of either z / 0 or H / 0 with the other
parameters (in the expression of z) fixed. Hence, near
the surface the wind profile differs very little from the
layer with a uniform temperature. Since jLj is on the
order of ;102 m (e.g., Irwin and Binkowski 1981) and
the depth of the dynamic sublayer may be comparable to
L (e.g., Carl et al. 1973), the classical interpretation of L
is potentially incompatible with the findings of Monin
and Yaglom (1971) and the view of Kader and Yaglom
(1990). Reconciliation of these opinions opens the possibility of an alternative interpretation of the physical
meaning of L, which we put forward below.
It appears to us that the classical interpretation of L
based on Obukhov’s reasoning (i.e., equating z / 0 to
z / 0 and/or H / 0) might be due to a mathematical
artifact. Note that according to the Monin–Obukhov
similarity theory, all state variables are determined
by the dimensionless parameter z in the similarity equations. Mathematically, various scenarios of limiting cases
should be examined under the condition of a given z.
That is, to understand the individual roles of z and H in
fm and fh in Eqs. (6) and (7) through z, one ought to first
hold z fixed, however small or large, and then find out
what will happen as z and/or H tend to the limits. When z
is fixed, z / 0 leads to H / ‘; meanwhile z / ‘ leads
to H / 0. Then we obtain an opposite conclusion to that
of the earlier authors: the buoyancy effect cannot be
ignored at small z, and mechanical mixing dominates the
turbulence away from the surface.
The alternative interpretation of L seems to be physically reasonable. It would be counterintuitive that the
buoyancy effect is negligible close to the source of heat
(i.e., the land surface where radiative energy is received)
and becomes more dominant away from the heat source.
This view appears to be consistent with the analysis of
fluid turbulence by Ozawa et al. (2001), who have shown
that the thickness of a molecular thermal (and viscous)
boundary layer is minimum; therefore the entire layer is
essentially dominated by turbulent thermal convection.
In addition, it has been known that the (turbulent) thermal convection corresponds to the largest temperature
gradient across the molecular thermal boundary layer
(Malkus 1954). Since a strong temperature gradient
exists near the surface owing to the difference between
the skin temperature and near-surface air temperature
and the largest temperature gradient occurs right next to
the surface, it would be difficult to imagine the buoyancy
FEBRUARY 2010
WANG AND BRAS
effect playing no role where the temperature gradient is
the strongest regardless of the effect of wind shear. It
would be equally counterintuitive that the buoyancy
effect dominates away from the surface (i.e., heat source)
where the source of kinetic energy driving the mechanical mixing is supplied by the mean flow. Furthermore,
the alternative interpretation of L is compatible with the
findings of Monin and Yaglom (1971) and the view of
Kader and Yaglom (1990) in the sense that the buoyancy-driven thermal turbulence is important through the
entire surface layer. We will return to this point later. It is
important to emphasize that the extremum solution, as
mathematical functions, is not affected by the interpretation of L, whichever it is.
The alternative interpretation of L, as well as the
extremum solution, suggests that the separation of the
‘‘forced convection regime’’ from the ‘‘free convection
regime’’ may be unimportant, even impossible, in modeling the turbulent transport since L could be a redundant
length scale parameter. A key assumption in the MOST is
that the turbulence is independent of the underlying
surface characteristics. Therefore, there should be no
other length scale than z itself because the effects of
buoyancy and wind shear are inseparably coupled. It is
true that L / ‘ when H / 0 with the other parameters
fixed. However, the existence of L results from the dimensionless combination of three parameters g/T0,
H/rcp, and u*. When H/rcp (or u*) is either undefined or
zero, L does not exist and the only length scale is z.
Hence, L is not continuous in H at H 5 0 (or in u* at
u* 5 0); that is, L (H 5 0) 6¼ limH/0 L 5 ‘. After all,
nonexistence of L is not the same thing as L / ‘. Under
this situation, L becomes undefined and, hence, loses its
physical significance.
Another common assumption in the MOST is the
height-invariant fluxes of momentum and heat. This
assumption makes L a length scale parameter independent of z. Yet height-invariant fluxes are rare, if at
all, in nature. Consider a typical scenario in which the
diurnal variation of temperature is ;20 K. The corresponding flux gradient is given by the energy balance
equation:
›H
›T
20
5 rcp
; 103 3
; 0.5 W m2 .
›z
›t
12 3 3600
Assuming the depth of a surface layer is O(50 m), the
variation of heat flux, DH, over the surface layer would
be DH ; 50 3 0.5 5 25 W m22, which is about 10% of
a typical value of H (;200 W m22). If the surface layer
is defined as the layer within which heat flux varies no
more than, say, 10% according to popular textbooks,
practically all surface layers should be treated as a layer
497
with variable fluxes rather than with constant fluxes.
Although use of the MOST in estimating fluxes has been
critically dependent on the assumption of constant
fluxes, this assumption plays no role in the extremum
solution proposed here.
The classical interpretation of the physical meaning of
L is closely associated with the assumption of the heightinvariant fluxes or L. According to the equation of turbulent kinetic energy (TKE), the relative magnitude of
thermal versus mechanical source of TKE is simply z/L.
It is true that the thermal effect on TKE is insignificant
for small z and dominant for large z when L is independent of z as assumed in the classical theories.
However, momentum and heat fluxes do vary in z for the
reasons explained in the previous paragraph so that L in
general varies with z. The extremum solution predicts
the ratio z/L equal to (2b)21 ; 0.1 for a stable layer and
2g1
2 ; 0.2 for an unstable layer, indicating that the
mechanical mixing is a much more effective production
mechanism than buoyancy throughout the entire surface
layer, although the production of TKE due to buoyancy
is twice as strong in the unstable layer than in the stable
layer. This is another reason that the classical view on
the meaning of L is open to debate.
6. Conclusions
The significance of the extremum solution of the
Monin–Obukhov similarity equation may be summarized
as 1) it is the only mathematically consistent and physically realistic solution of the Monin–Obukhov similarity
equations; 2) it has overcome some technical difficulties
(e.g., nonuniqueness and nonconvergence) in applying
the MOST in modeling of turbulent transport in the ASL;
3) it opens a possibility of simplifying the MOST formalism by replacing the two empirical stability functions
by some empirical constants; 4) it unifies the asymptotic
solutions of the ASL derived from various arguments; 5)
it has a solid foundation built on the modern nonequilibrium thermodynamics; and 6) it can play a crucial
role in successful modeling of the surface heat fluxes
based on the emerging theory of maximum entropy
production. Our analysis also raises a doubt on the
classical interpretation of the Obukhov length, arguably
due to a mathematical artifact. Resolution of the issue
may offer new opportunities in improving atmospheric
turbulence models.
Acknowledgments. This work was supported by ARO
Grant W911NF-07-1-0126, NSF Grant EAR-0309594, and
NASA Grants NAG5-3726 (TRMM) and NNG05GA17G.
The equipment for the field experiment at Lucky Hill
498
JOURNAL OF THE ATMOSPHERIC SCIENCES
was provided by the ARO project W911NF-06-1-0224. We
thank David Goodrich and John Smith of USDA-ARS for
their support during the field experiment involving graduate students at MIT, Ryan Knox and Gajan Sivandran.
This study was initiated by an inspiring conversation of the
first author with Guido Salvucci of Boston University. We
sincerely thank Gabriel Katul of Duke University for
providing the field data from the Owens Lake site. We
are grateful to Christine Sherratt at Lindgren Library of
MIT for her assistance in obtaining the English translation of the classical papers of Monin and Obukhov.
Insightful comments by two anonymous reviewers have
led to clarification of several important issues and significantly improved the quality of this paper.
APPENDIX
Based on the theoretical framework of maximum entropy production (Dewar 2005), Wang and Bras (2009)
proposed a model of H and ground heat flux G over a dry
land surface. Maximizing a dissipation function (or entropy production function) under the constraint of conservation of energy results in the following expression:
Is
HjHj1/6 .
I0
(A1)
Combining with energy balance equation,
G 1 H 5 Rn ,
(A2)
for a given net radiation input Rn leads to a solution of H
and G (as functions of the given Rn). In Eq. (A1) Is is the
thermal inertia parameter for heat conduction in the soil
and I0 the coefficient in the ‘‘thermal inertia’’ for turbulent heat transfer in the boundary layer, Ia, expressed
in terms of the eddy diffusivity KH given in Eq. (30):
I a [ rCp
qffiffiffiffiffiffiffiffi
K H 5 I 0 jHj1/6 ,
(A3)
where the extremum solution was used to substitute u* in
KH by H. Here I0 is expressed in terms of the coefficients
in the stability functions, identified by Businger et al.
(1971),
qffiffiffiffiffiffiffiffiffiffiffi
kzg
I 0 5 rCp C1 kz C2
rCp T 0
where
8pffiffiffi
3
>
>
< ,
a
C1 5
2
>
>
:
,
(1 1 2a)
unstable
stable
and
(g
2
,
C2 5 2
2b,
unstable
stable.
Other parameters are given in Table 1.
REFERENCES
A MEP Model of Heat Fluxes over a Dry
Land Surface
G5
VOLUME 67
!1/6
,
(A4)
Arya, S. P., 1988: Introduction to Micrometeorology. Academic
Press, 307 pp.
Beljaars, A. C. M., and A. A. M. Holtslag, 1991: Flux parameterization over land surfaces for atmospheric models. J. Appl.
Meteor., 30, 327–341.
Bridgman, P. W., 1931: Dimensional Analysis. Yale University
Press, 113 pp.
Brutsaert, W., 2005: Hydrology: An Introduction. Cambridge University Press, 605 pp.
Buckingham, E., 1914: On physically similar systems: Illustrations
of the use of dimensional equations. Phys. Rev., 4, 345–376.
Businger, J. A., J. Wyngaard, Y. Izumi, and E. F. Bradley, 1971:
Flux-profile relationships in the atmospheric surface layer.
J. Atmos. Sci., 28, 181–189.
Busse, F. H., 1978: The optimum theory of turbulence. Adv. Appl.
Mech., 18, 77–121.
Carl, D. M., T. C. Tarbell, and H. A. Panofsky, 1973: Profiles of
wind and temperature from towers over homogeneous terrain.
J. Atmos. Sci., 30, 788–794.
Cheng, Y. M., M. B. Parlange, and W. Brutsaert, 2005: Pathology
of Monin–Obukhov similarity in the stable boundary layer.
J. Geophys. Res., 110, D06101, doi:10.1029/2004JD004923.
Dewar, R. C., 2003: Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized
criticality in non-equilibrium stationary states. J. Phys., 36,
631–641.
——, 2005: Maximum entropy production and the fluctuation theorem. J. Phys., 38, L371–L381, doi:10.1088/0305-4470/38/21/L01.
Dyer, A. J., 1964: The flux-gradient relation for turbulent heat
transfer in the lower atmosphere. Quart. J. Roy. Meteor. Soc.,
90, 151–157.
——, 1967: The turbulent transport of heat and water vapour in an
unstable atmosphere. Quart. J. Roy. Meteor. Soc., 93, 501–508.
Elliott, W. P., 1964: The height variation of vertical heat flux near
the ground. Quart. J. Roy. Meteor. Soc., 90, 260–265.
Goodman, L. E., and W. H. Warner, 2001: Dynamics. Dover,
617 pp.
Handorf, D., T. Foken, and C. Kottmeier, 1999: The stable atmospheric boundary layer over an Antarctic ice sheet. Bound.Layer Meteor., 91, 165–189.
Irwin, J. S., and F. S. Binkowski, 1981: Estimation of the Monin–
Obukhov scaling length using on-site instrumentation. Atmos.
Environ., 15, 1091–1094.
FEBRUARY 2010
WANG AND BRAS
Jaynes, E. T., 1957: Information theory and statistical mechanics.
Phys. Rev., 106, 620–630.
——, 2003: Probability Theory: The Logic of Science. Cambridge
University Press, 758 pp.
Johansson, C., A. Smedman, U. Högström, J. G. Brasseur, and
S. Khanna, 2001: Critical test of the validity of Monin–Obukhov
similarity during convective conditions. J. Atmos. Sci., 58,
1549–1566.
Kader, B. A., and A. M. Yaglom, 1990: Mean fields and fluctuation
moments in unstably stratified turbulent boundary layers.
J. Fluid Mech., 212, 637–662.
Katul, G. G., 1994: A model for sensible heat flux probability
density function for near-neutral and slightly-stable atmospheric flows. Bound.-Layer Meteor., 71, 1–20.
Kleidon, A., K. Fraedrich, E. Kirk, and F. Lunkeit, 2006: Maximum
entropy production and the strength of boundary layer exchange in an atmospheric general circulation model. Geophys.
Res. Lett., 33, L06706, doi:10.1029/2005GL025373.
Kondepudi, D., and I. Prigogine, 1998: Modern Thermodynamics.
Wiley, 486 pp.
Lanczos, C., 1970: The Variational Principles of Mechanics. Dover,
418 pp.
Lorenz, R. D., and C. P. McKay, 2003: A simple expression for
vertical convective fluxes in planetary atmospheres. Icarus,
165, 407–413.
Malkus, W. V. R., 1954: The heat transport and spectrum of thermal turbulence. Proc. Roy. Soc. London, 225A, 196–212.
Monin, A. S., and A. M. Obukhov, 1954: Basic turbulence mixing
laws in the atmospheric surface layer. Tr. Inst. Teor. Geofiz.
Akad. Nauk. SSSR, 24, 163–187. (English translation available
499
in V. N. Bespalyi, Ed., 2001: Turbulence and Atmospheric
Dynamics, J. L. Lumley, 164–194.)
——, and A. M. Yaglom, 1971: Statistical Fluid Mechanics: Mechanics
of Turbulence. Vol. 1, The MIT Press, 782 pp.
Obukhov, A. M., 1946: Turbulence in an atmosphere with a nonuniform temperature. Tr. Inst. Teor. Geofiz. Akad. Nauk.
SSSR, 1, 95–115. (English translation available in Obukhov,
A. M., 1971: Turbulence in an atmosphere with a non-uniform
temperature. Boundary-Layer Meteor., 2, 7–29.)
Ozawa, H., S. Shimokawa, and H. Sakuma, 2001: Thermodynamics
of fluid turbulence: A unified approach to the maximum
transport properties. Phys. Rev. E, 64, 026303, doi:10.1103/
PhysRevE.64.026303.
Priestley, C. H. B., 1955: Free and forced convection in the atmosphere near the ground. Quart. J. Roy. Meteor. Soc., 81, 139–143.
Shannon, C. E., and W. Weaver, 1949: The Mathematical Theory of
Communication. University of Illinois Press, 144 pp.
Taylor, R. J., 1956: Some measurements of heat flux at large negative
Richardson number. Quart. J. Roy. Meteor. Soc., 82, 89–91.
——, 1959: Similarity theory in the relation between fluxes and
gradients in the lower atmosphere. Quart. J. Roy. Meteor. Soc.,
85, 67–78.
Townsend, A. A., 1962: Natural convection in the earth’s boundary
layer. Quart. J. Roy. Meteor. Soc., 88, 51–56.
Wang, J., and R. L. Bras, 2009: A model of surface heat fluxes based
on the theory of maximum entropy production. Water Resour.
Res., 45, W11422, doi:10.1029/2009WR007900.
——, G. D. Salvucci, and R. L. Bras, 2004: An extremum principle
of evaporation. Water Resour. Res., 40, W09303, doi:10.1029/
2004WR003087.